Example: Algebraic Multiplicity vs Geometric Multiplicity

Question

Is there a simple example of a matrix having an eigenvalue whose geometric multiplicity is strictly smaller than its algebraic multiplicity?

Answer 1

Consider $$ A = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} $$ As $\chi_A(t) = t^2$, $0$ has algebraic multiplicity $2$, but geometric multiplicity $1$.

Answer 2

The matrices having at least one eigenvalue with geometric multiplicity smaller than its algebraic multiplicity are exactly the non-diagonalizable ones.
The comments and answer illustrate this: non-zero nilpotent matrices are examples of non-diagonalizable matrices (but there are many other examples !).